Method of measuring warpage of rear surface of substrate

ABSTRACT

A method of measuring warpage of a rear surface of a substrate includes a substrate detection step, a best fit plane calculation step, and a warpage calculation step. Further, the method of measuring warpage of a rear surface of a substrate can further includes after the substrate detection step and before the best fit plane calculation step: a noise removal step and an outer peripheral portion removal step; the outer peripheral portion removal step and a smoothing step; or the noise removal step, the outer peripheral portion removal step, and the smoothing step. Thereby, a method of measuring warpage of a rear surface with a high surface roughness of a substrate can be provided.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a method of measuring warpage of a rearsurface of a substrate used in a semiconductor device or the like, andmore specifically, to a method of measuring warpage of a rear surface (asurface opposite to a crystal growth surface; hereinafter the sameapplies) of a substrate using a laser displacement meter.

2. Description of the Background Art

In a substrate used in a semiconductor device or the like, it isnecessary to form one or more semiconductor layers having good qualityon the substrate in order to obtain a semiconductor device havingexcellent properties. Accordingly, the substrate is required to have acrystal growth surface with reduced warpage and surface roughness. Thewarpage of the crystal growth surface can be measured by a flatnesstester employing optical interferometry, and the surface roughness ofthe crystal growth surface can be measured by a 3D-SEM(three-dimensional scanning electron microscope; hereinafter the sameapplies) or the like (see for example “Superprecision Wafer SurfaceControl Technology” by Yoshiaki Matushita et al., the first edition,Science Forum Inc., Feb. 28, 2000, pages 258-264 and 272-278 (Non-PatentDocument 1)).

In order to form one or more semiconductor layers having good quality onthe substrate, the substrate is required to not only have a crystalgrowth surface with reduced warpage and surface roughness, but also havea rear surface with reduced warpage and surface roughness. If the rearsurface has large warpage and surface roughness, this causes an increasein a gap portion formed between the rear surface of the substrate and asusceptor (meaning a table on which a substrate is disposed; hereinafterthe same applies) when a semiconductor layer is formed on the crystalgrowth surface of the substrate. As a result, heat transferred from thesusceptor to the substrate is unevenly distributed, and thesemiconductor layer cannot be formed evenly and stably on the crystalgrowth surface of the substrate. Consequently, a semiconductor layerhaving good quality cannot be formed.

Consequently, in order to prepare a substrate suitable for fabricationof a semiconductor device, it is necessary to evaluate not only warpageand surface roughness of a crystal growth surface of a substrate butalso warpage and surface roughness of a rear surface of the substrate.The surface roughness of the rear surface can be measured by a 3D-SEM orthe like.

However, the rear surface has a surface roughness greater than that ofthe crystal growth surface, and it often has a surface roughness Ra ofnot less than 50 nm. Accordingly, it is difficult to measure the warpageof the rear surface by a flatness tester employing opticalinterferometry. Further, since the flatness tester employing opticalinterferometry cannot obtain a reflected beam, it cannot provide dataanalysis.

Therefore, there has been a strong need to develop a method of measuringwarpage of a rear surface of a substrate in order to fabricate asemiconductor device having excellent properties.

SUMMARY OF THE INVENTION

One object of the present invention is to provide a method of measuringwarpage of a rear surface of a substrate.

The present invention is a method of measuring warpage of a rear surfaceopposite to a crystal growth surface of a substrate using a laserdisplacement meter, the substrate being disposed on a substrate supporttable, including: a substrate detection step detecting a plurality ofdisplacement values respectively corresponding to a plurality ofmeasurement points on the rear surface of the substrate using the laserdisplacement meter; a noise removal step removing noise contained in theplurality of displacement values; an outer peripheral portion removalstep calculating a plurality of displacement values for calculation byremoving from the plurality of displacement values those respectivelycorresponding to the measurement points in an outer peripheral portionof the substrate; a smoothing step smoothing the plurality ofdisplacement values for calculation to calculate a warped surface; abest fit plane calculation step calculating a best fit plane having theminimum distance to the warped surface; and a warpage calculation stepcalculating as warpage a sum of a distance from the best fit plane to apoint represented by the greatest displacement value of the warpedsurface on one side with respect to the best fit plane and a distancefrom the best fit plane to a point represented by the greatestdisplacement value of the warped surface on the other side with respectto the best fit plane.

In the method of measuring warpage of a rear surface of a substrate inaccordance with the present invention, the substrate can be disposed onthe substrate support table having three supporting portions such thatthe crystal growth surface of the substrate is supported by the threesupporting portions. Further, the substrate detection step can beperformed by measuring distances between the laser displacement meterand the plurality of measurement points on the rear surface by a laserfocus technique while moving the substrate support table on which thesubstrate is disposed in a two-dimensional direction in a stepwisefashion. Further, the noise removal step can be performed using a medianfilter. Furthermore, the smoothing step can be performed using aGaussian filter. Further, the best fit plane calculation step can beperformed by calculating the best fit plane to minimize a sum of squaresof every distance between the best fit plane and each point representedby each of the plurality of displacement values for calculationsubjected to the smoothing.

Furthermore, in the method of measuring warpage of a rear surface of asubstrate in accordance with the present invention, an optimizationcycle including the smoothing step, the best fit plane calculation step,and the warpage calculation step can be repeated one or more times.Further, at least one noise removal step can be included in an intervalbetween the repeated optimization cycles, or after the smoothing step inthe optimization cycle.

Further, the present invention is a method of measuring warpage of arear surface opposite to a crystal growth surface of a substrate using alaser displacement meter, the substrate being disposed on a substratesupport table, including: a substrate detection step detecting aplurality of displacement values respectively corresponding to aplurality of measurement points on the rear surface of the substrateusing the laser displacement meter; a best fit plane calculation stepcalculating a best fit plane having the minimum distance to a pluralityof points respectively represented by the plurality of displacementvalues; and a warpage calculation step calculating as warpage a sum of adistance from the best fit plane to a point represented by the greatestdisplacement value on one side with respect to the best fit plane and adistance from the best fit plane to a point represented by the greatestdisplacement value on the other side with respect to the best fit plane,in the plurality of points respectively represented by the plurality ofdisplacement values.

Further, the method of measuring warpage of a rear surface of asubstrate in accordance with the present invention can further includeafter the substrate detection step and before the best fit planecalculation step: a noise removal step removing noise contained in theplurality of displacement values; and an outer peripheral portionremoval step calculating a plurality of displacement values forcalculation by removing from the plurality of displacement values thoserespectively corresponding to the measurement points in an outerperipheral portion of the substrate, and can use the plurality ofdisplacement values for calculation as the plurality of displacementvalues in the best fit plane calculation step and the warpagecalculation step. On this occasion, the noise removal step can beperformed using a median filter.

Further, the method of measuring warpage of a rear surface of asubstrate in accordance with the present invention can further includeafter the substrate detection step and before the best fit planecalculation step: an outer peripheral portion removal step calculating aplurality of displacement values for calculation by removing from theplurality of displacement values those respectively corresponding to themeasurement points in an outer peripheral portion of the substrate; anda smoothing step smoothing the plurality of displacement values forcalculation to calculate a warped surface, and can use the plurality ofpoints respectively represented by the plurality of displacement valuesfor calculation subjected to smoothing on the warped surface as theplurality of points respectively represented by the plurality ofdisplacement values in the best fit plane calculation step and thewarpage calculation step. On this occasion, the smoothing step can beperformed using a Gaussian filter.

Further, in the method of measuring warpage of a rear surface of asubstrate in accordance with the present invention, the substrate can bedisposed on the substrate support table having three supporting portionssuch that the crystal growth surface of the substrate is supported bythe three supporting portions. Further, the substrate detection step canbe performed by measuring distances between the laser displacement meterand the plurality of measurement points on the rear surface by a laserfocus technique while moving the substrate support table on which thesubstrate is disposed in a two-dimensional direction in a stepwisefashion.

According to the present invention, a method of measuring warpage of arear surface of a substrate can be provided.

The foregoing and other objects, features, aspects and advantages of thepresent invention will become more apparent from the following detaileddescription of the present invention when taken in conjunction with theaccompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a flow chart illustrating an example of a method of measuringwarpage of a rear surface of a substrate in accordance with the presentinvention.

FIG. 2 is a schematic view showing a measuring apparatus used in themethod of measuring the warpage of the rear surface of the substrate inaccordance with the present invention.

FIG. 3 is a schematic plan view showing a plurality of measurementpoints in the method of measuring the warpage of the rear surface of thesubstrate in accordance with the present invention.

FIG. 4 is a schematic plan view showing an arrangement of the pluralityof measurement points.

FIG. 5A is a schematic view of a kernel for an 8-neighborhood Gaussianfilter illustrating positions at which Gaussian functions f(x, y)serving as coefficients are arranged.

FIG. 5B is a schematic view of a kernel for an 8-neighborhood Gaussianfilter illustrating an arrangement of coefficients with σ=5 beforenormalization.

FIG. 5C is a schematic view of a kernel for an 8-neighborhood Gaussianfilter illustrating an arrangement of coefficients with σ=5 afternormalization.

FIGS. 6A and 6B are schematic views showing a warpage calculation stepin the method of measuring the warpage of the rear surface of thesubstrate in accordance with the present invention.

FIG. 7 is a flow chart illustrating another example of the method ofmeasuring the warpage of the rear surface of the substrate in accordancewith the present invention.

FIG. 8 is a flow chart illustrating still another example of the methodof measuring the warpage of the rear surface of the substrate inaccordance with the present invention.

FIG. 9 is a flow chart illustrating still another example of the methodof measuring the warpage of the rear surface of the substrate inaccordance with the present invention.

FIGS. 10A and 10B are schematic views showing another example of thewarpage calculation step in the method of measuring the warpage of therear surface of the substrate in accordance with the present invention.

FIGS. 11A and 11B are schematic views showing still another example ofthe warpage calculation step in the method of measuring the warpage ofthe rear surface of the substrate in accordance with the presentinvention.

DESCRIPTION OF THE PREFERRED EMBODIMENTS First Embodiment

Referring to FIG. 1, one embodiment of a method of measuring warpage ofa rear surface of a substrate in accordance with the present inventionis a method of measuring warpage of a rear surface opposite to a crystalgrowth surface of a substrate using a laser displacement meter, thesubstrate being disposed on a substrate support table. The methodincludes: a substrate detection step S1 detecting a plurality ofdisplacement values respectively corresponding to a plurality ofmeasurement points on the rear surface of the substrate using the laserdisplacement meter; a noise removal step S2 removing noise contained inthe plurality of displacement values; an outer peripheral portionremoval step S3 calculating a plurality of displacement values forcalculation by removing from the plurality of displacement values thoserespectively corresponding to the measurement points in an outerperipheral portion of the substrate; a smoothing step S4 smoothing theplurality of displacement values for calculation to calculate a warpedsurface; a best fit plane calculation step S5 calculating a best fitplane having the minimum distance to the warped surface; and a warpagecalculation step S6 calculating as warpage a sum of a distance from thebest fit plane to a point represented by the greatest displacement valueof the warped surface on one side with respect to the best fit plane anda distance from the best fit plane to a point represented by thegreatest displacement value of the warped surface on the other side withrespect to the best fit plane. With the measurement method describedabove, even for a substrate having a rear surface with a high surfaceroughness (for example, with a surface roughness Ra of not less than 50nm), the warpage of the rear surface of the substrate can be measured.It is to be noted that surface roughness Ra is a value obtained bysampling a portion having a reference length from a roughness curve in adirection of its mean line, summing up absolute values of deviationsfrom a mean line of the sampled portion to a measurement curve, andcalculating an average for the reference length.

Further, in FIG. 1, the step surrounded with a solid frame is anindispensable step, and the step surrounded with a dashed frame is anarbitrary step. While the present embodiment and FIG. 1 describe thecase where the outer peripheral portion removal step S3 is performedafter the noise removal step S2, these steps may be performed in inverseorder.

Turning to FIG. 2, a laser displacement meter 15 is an apparatusmeasuring a displacement of a rear surface 10 r of a substrate 10 byapplying a laser beam 21 on rear surface 10 r of substrate 10. There isno particular limitation on the type of the laser, and for example a redcolor semiconductor laser having a wavelength of 670 nm is used. Thereis no particular limitation on the measuring technique, and for examplea laser focus technique is used. Although a laser displacement meteremploying the laser focus technique has a lower measuring accuracy thana flatness tester employing optical interferometry, it can measure arough rear surface with surface roughness Ra of not less than 50 nm. Themeasuring accuracy of the laser displacement meter can be improved byusing a blue color semiconductor laser having a wavelength shorter thanthat of a red color semiconductor laser. Further, unlike the flatnesstester employing optical interferometry, the laser displacement meteremploying the laser focus technique can obtain a reflected beam 21 r,and thus it can analyze and process a displacement value.

Referring to FIGS. 2 and 3, substrate 10 is disposed on a substratesupport table 12. Although there is no particular limitation on how todispose substrate 10 on substrate support table 12, substrate 10 ispreferably disposed on substrate support table 12 having threesupporting portions 12 h such that a crystal growth surface 10 c ofsubstrate 10 is supported by the three supporting portions 12 h.Supporting an outer peripheral portion of crystal growth surface 10 c ofsubstrate 10 only by the three supporting portions 12 h can minimizedamage to crystal growth surface 10 c during the measurement of warpage.Further, even when substrate 10 is inclined while being supported by theabove three portions, the inclination of substrate 10 can be compensatedfor by calculating a best fit plane having the minimum distance to awarped surface (meaning a curved surface indicating warpage of a rearsurface; hereinafter the same applies), and calculating a distance fromthe best fit plane to the warped surface.

Referring to FIGS. 1 to 3, although there is no particular limitation onthe substrate detection step S1, the step can be performed by measuringa distance L between laser displacement meter 15 and rear surface 10 rof substrate 10 while moving substrate 10 in a two-dimensional direction(meaning an X direction and a Y direction in FIG. 3; hereinafter thesame applies) in a stepwise fashion. The stepwise movement of substrate10 in the two-dimensional direction can be performed by moving a drivingportion 13 coupling substrate support table 12 to a driving unit 14 inthe two-dimensional direction in a stepwise fashion. Driving unit 14 iscontrolled by a position controlling unit 16.

On this occasion, position data in the two-dimensional direction of ameasurement point 100 p (an arbitrarily specified measurement point)irradiated with laser beam 21 among the plurality of measurement pointson the rear surface of the substrate is collected to a data analysisunit 18 via position controlling unit 16. Here, an arrow 22 indicates adirection in which the position data is transmitted.

While there is no particular limitation on how to measure distance L, itcan for example be measured by the laser focus technique. The laserfocus technique will now be described below. An incident beam 21 iemitted from a light source in laser displacement meter 15 is applied toarbitrarily specified measurement point 100 p on rear surface 10 r ofsubstrate 10 via an objective lens (not shown) moved up and down at ahigh speed within laser displacement meter 15 by means of a tuning fork.Reflected beam 21 r from arbitrarily specified measurement point 100 ppasses through a pin hole (not shown) in laser displacement meter 15 andreaches a light receiving element (not shown). According to the confocalprinciple, when incident beam 21 i is focused on arbitrarily specifiedmeasurement point 100 p on rear surface 10 r of substrate 10, reflectedbeam 21 r is focused into one point at a position of the pin hole andenters the light receiving element. By measuring a position of theturning fork on this occasion with a sensor (not shown), distance Lbetween laser displacement meter 15 and arbitrarily specifiedmeasurement point 100 p on rear surface 10 r of substrate 10 can bemeasured. With this manner, a displacement value z_((a, b)) (meaning adisplacement value in a Z direction; hereinafter the same applies) ofarbitrarily specified measurement point 100 p on rear surface 10 r ofsubstrate 10 can be measured.

On this occasion, displacement value data of arbitrarily specifiedmeasurement point 100 p among a plurality of measurement points 10 p onrear surface 10 r of substrate 10 is collected to data analysis unit 18via a laser displacement meter controlling unit 17. Here, an arrow 23indicates a direction in which the displacement value data istransmitted.

Next, the above measurement is performed after the substrate is moved ina stepwise fashion (for example in the X direction or the Y direction ata constant pitch P) as shown in FIGS. 2 and 3, and thus the positiondata in the two-dimensional direction (the X direction and the Ydirection) and the displacement value data in the Z direction of ameasurement point adjacent to arbitrarily specified measurement point100 p at pitch P can be obtained. Through repeating the above operation,the position data in the two-dimensional direction (the X direction andthe Y direction) and the displacement value data in the Z direction ofeach of the plurality of measurement points 10 p on rear surface 10 r ofsubstrate 10 can be obtained. The position data in the two-dimensionaldirection (the X direction and the Y direction) and the displacementvalue data in the Z direction obtained as described above are collectedto data analysis unit 18.

As shown in FIG. 3, when substrate 10 of a circular shape is moved in astepwise fashion at constant pitch P in the two-dimensional direction(the X direction and the Y direction), there may be a case where thelaser beam is applied to substrate support table 12 instead of rearsurface 10 r of substrate 10. As shown in FIG. 3, when substrate 10 isdisposed in a concave portion of substrate support table 12, there maybe a measurement point 120 a on a surface 12 a of a non-concave portionof substrate support table 12, and a measurement point 120 b on asurface 12 b of the concave portion of substrate support table 12.

In such a case, referring to FIG. 2, the plurality of displacementvalues respectively corresponding to the plurality of measurement points10 p on rear surface 10 r of substrate 10 can be detected withmeasurement points 120 a and 120 b removed as described below.Specifically, measurement points 120 a and 120 b can be removed bydetecting only arbitrarily specified measurement point 100 p which hasdistance L to laser displacement meter 15 satisfying the relationLa<L<Lb, where La is a distance between laser displacement meter 15 andsurface 12 a of the non-concave portion of substrate support table 12,and Lb is a distance between laser displacement meter 15 and surface 12b of the concave portion of substrate support table 12. Consequently,the plurality of displacement values respectively corresponding to theplurality of measurement points 10 p on rear surface 10 r of substrate10 can be obtained.

In the present embodiment, although there is no particular limitation onthe noise removal step S2 as long as it removes noise contained in theplurality of displacement values, it is preferable to use a medianfilter for the step. Referring to FIG. 4, a median filter is a filterreplacing a displacement value z_((a, b)) (meaning a displacement valuecorresponding to arbitrarily specified measurement point 100 p;hereinafter the same applies) specified arbitrarily among the pluralityof displacement values (meaning the plurality of displacement valuesrespectively corresponding to the plurality of measurement points 10 pon rear surface 10 r of substrate 10; hereinafter the same applies) by amedian obtained when arranging the displacement value z_((a, b)) and aplurality of displacement values z_((a−1, b+1)), z_((a−1, b)),z_((a−1, b−1)), z_((a, b+1)), z_((a, b−1)), z_((a+1, b+1)),z_((a+1, b)), and z_((a+1, b−1)) neighboring the displacement valuez_((a, b)) (meaning displacement values respectively corresponding to aplurality of measurement points 101 p, 102 p, 103 p, 104 p, 105 p, 106p, 107 p, 108 p neighboring arbitrarily specified measurement point 100p; hereinafter the same applies) in increasing or decreasing order. InFIG. 4, the displacement value z_((a, b)) and the plurality ofdisplacement values z_((a−1, b+1)), z_((a−1, b)), z_((a−1, b−1)),z_((a, b+1)), z_((a, b−1)), z_((a+1, b+1)), z_((a+1, b)), andz_((a+1, b−1)) neighboring the displacement value z_((a, b)) arearranged in the two-dimensional direction (the X direction and the Ydirection) at constant pitch P.

Although FIG. 4 shows eight displacement values z_((a−1, b+1)),z_((a−1, b)), z_((a−1, b−1)), z_((a, b+1)), z_((a, b−1)),z_((a+1, b+1)), z_((a+1, b)), and z_((a+1, b−1)) neighboring andsurrounding the arbitrarily specified displacement value as theplurality of neighboring displacement values (such a median filter iscalled an 8-neighborhood median filter), the number of the plurality ofneighboring measuring points is not limited to eight. For example, 24measuring points neighboring a displacement value can also be used (sucha median filter is called an 24-neighborhood median filter).

In the present embodiment, there is no particular limitation on theouter peripheral portion removal step S3 as long as it calculates aplurality of displacement values for calculation by removing from theplurality of displacement values those respectively corresponding to themeasurement points in an outer peripheral portion of the substrate. Whenusing an 8-neighborhood median filter in the noise removal step S2,however, referring to FIG. 3, it is preferable to remove displacementvalues respectively corresponding to at least two measurement points 111p and 112 p inward from an outer periphery 10 e, as the displacementvalues respectively corresponding to the measurement points in the outerperipheral portion of substrate 10, from the plurality of displacementvalues.

This is because, when an 8-neighborhood median filter is used in thenoise removal step S2, referring to FIG. 3, at least one of eightdisplacement values neighboring a displacement value at a position oneor two points inward from outer periphery 10 e of substrate 10 is adisplacement value of surface 12 a of the non-concave portion or surface12 b of the concave portion of substrate support table 12, and thus theabove noise removal step fails to remove noise. With this manner, thedisplacement values respectively corresponding to the measurement pointsin the outer peripheral portion of the substrate are removed from theplurality of displacement values, and the plurality of displacementvalues for calculation is obtained.

In the present embodiment, referring to FIGS. 6A and 6B, although thereis no limitation on the smoothing step S4 as long as it smoothes theplurality of displacement values for calculation to calculate a warpedsurface 30, it is preferable to use a Gaussian filter for the step. AGaussian filter is a filter replacing a displacement value z_((a, b))specified arbitrarily among the plurality of displacement values forcalculation by a weighted average value z′_((a, b)) of the displacementvalue z_((a, b)) and the plurality of displacement valuesz_((a−1, b+1)), z_((a−1, b)), z_((a−1, b−1)), z_((a, b+1)),z_((a, b−1)), z_((a+1, b+1)), z_((a+1, b)), and z_((a+1, b−1))neighboring the displacement value z_((a, b)), using a Gaussian functionf(x, y) as a weighting factor. With the smoothing described above, evenfor a rear surface having a high surface roughness (for example, asurface roughness Ra of not less than 50 nm), the warpage of the rearsurface can be measured.

The two-dimensional Gaussian function f(x, y) is expressed by thefollowing equation (1):

$\begin{matrix}\begin{matrix}{{f\left( {x,y} \right)} = {\frac{1}{N^{2}}\exp \left\{ {- \frac{\left( {x - a} \right)^{2} + \left( {y - b} \right)^{2}}{2\sigma^{2}}} \right\}}} \\{= {\frac{1}{N^{2}}\exp {\left\{ {- \frac{\left( {x - a} \right)^{2}}{2\sigma^{2}}} \right\} \cdot \exp}\left\{ {- \frac{\left( {y - b} \right)^{2}}{2\sigma^{2}}} \right\}}}\end{matrix} & (1)\end{matrix}$

where a and b are coordinate values of an arbitrarily specifiedmeasurement point in the X direction and the Y direction, respectively,σ is a standard deviation (σ² is a dispersion), and N is a normalizationconstant.

As can be seen from equation (1), the greater the distance between ameasurement point (x, y) and an arbitrarily specified measurement point(a, b) is, the smaller and less weighted the value of f(x, y) becomes.Further, the greater the value of σ is, the smaller the difference inweighting resulting from the difference in the distance between themeasurement point (x, y) and the arbitrarily specified measurement point(a, b) becomes.

Although eight displacement values z_((a−1, b+1)), z_((a−1, b)),z_((a−1, b−1)), z_((a, b+1)), z_((a, b−1)), z_((a+1, b+1)),z_((a+1, b)), and z_((a+1, b−1)) neighboring and surrounding anarbitrarily specified displacement value are used in the above as theplurality of neighboring displacement values (such a Gaussian filter iscalled an 8-neighborhood Gaussian filter), the number of the pluralityof neighboring displacement values is not limited to eight. For example,24 displacement values neighboring a displacement value can also be used(such a Gaussian filter is called an 24-neighborhood Gaussian filter).

Using an 8-neighborhood Gaussian filter specifically means replacing thedisplacement value z_((a, b)) specified arbitrarily by the weightedaverage value z′_((a, b)) obtained by weighted averaging of theplurality of displacement values z_((a−1, b+1)), z_((a−1, b)),z_((a−1, b−1)), z_((a, b+1)) _(, z) _((a, b)), z_((a, b−1)),z_((a+1, b+1)), z_((a+1, b)), and z_((a+1, b−1)) shown in FIG. 4, witheach of the values weighted by the Gaussian function f(x, y) (wherex=a−1, a, a+1; y=b−1, b, b+1) as a coefficient shown in a kernel(meaning a matrix of coefficients of a filter for displacement values;hereinafter the same applies) of FIG. 5A. Specifically, it meansobtaining the value z′_((a, b)) according to the following equation (2):

$\begin{matrix}{\begin{matrix}{z_{({a,b})}^{\prime} = {\sum\limits_{x = {a - 1}}^{a + 1}{\sum\limits_{y = {b - 1}}^{b + 1}{{f\left( {x,y} \right)} \cdot z_{({x,y})}}}}} \\{= {\frac{1}{N^{2}}{\sum\limits_{x = {a - 1}}^{a + 1}{\sum\limits_{y = {b - 1}}^{b + 1}{\exp {\left\{ {- \frac{\left( {x - a} \right)^{2} + \left( {y - b} \right)^{2}}{2\sigma^{2}}} \right\} \cdot z_{({x,y})}}}}}}} \\{= {\frac{1}{N^{2}}{\sum\limits_{x = {a - 1}}^{a + 1}{\sum\limits_{y = {b - 1}}^{b + 1}{\exp {\left\{ {- \frac{\left( {x - a} \right)^{2}}{2\sigma^{2}}} \right\} \cdot \exp}{\left\{ {- \frac{\left( {y - b} \right)^{2}}{2\sigma^{2}}} \right\} \cdot z_{({x,y})}}}}}}}\end{matrix}\left( {{{where}\mspace{14mu} N^{2}} = {\sum\limits_{x = {a - 1}}^{a + 1}{\sum\limits_{y = {b - 1}}^{b + 1}{\exp {\left\{ {- \frac{\left( {x - a} \right)^{2}}{2\sigma^{2}}} \right\} \cdot \exp}\left\{ {- \frac{\left( {y - b} \right)^{2}}{2\sigma^{2}}} \right\}}}}} \right)} & (2)\end{matrix}$

The Gaussian function f(x, y) serving as a coefficient of the Gaussianfilter is determined by the distance from the measurement point (a, b)of the arbitrarily specified displacement value to the measurement point(x, y) and by standard deviation σ. For example, FIG. 5B illustrates anarrangement of values of coefficients f(x, y) of an 8-neighborhoodGaussian filter with σ=5 before normalization, and FIG. 5C illustratesan arrangement of values of coefficients f(x, y) of an 8-neighborhoodGaussian filter with σ=5 after normalization. Normalization meanscorrecting coefficients f(x, y) of a Gaussian filter such that a sum ofthe coefficients f(x, y) is 1, while maintaining ratios between thecoefficients f(x, y).

Since the Gaussian function is independent of each of the X directionand the Y direction as can be seen from equation (1), equation (2) canbe separated into the form of a sum in the X direction and a sum in theY direction as shown in the following equations (3) and (4),respectively. This supports that applying smoothing with a Gaussianfilter to a method of performing data processing on a plurality ofdisplacement values obtained by firstly sampling a displacement value inthe X direction and subsequently sampling a displacement value in the Ydirection is mathematically reasonable.

$\begin{matrix}{z_{({a,y})}^{\prime} = {\frac{1}{N}{\sum\limits_{x = {a - 1}}^{a + 1}{\exp {\left\{ {- \frac{\left( {x - a} \right)^{2}}{2\sigma^{2}}} \right\} \cdot z_{({x,y})}}}}}} & (3) \\{z_{({a,b})}^{\prime} = {\frac{1}{N}{\sum\limits_{y = {b - 1}}^{b + 1}{\exp {\left\{ {- \frac{\left( {y - b} \right)^{2}}{2\sigma^{2}}} \right\} \cdot z_{({a,y})}^{\prime}}}}}} & (4)\end{matrix}$

Further, since the Gaussian function is symmetrical with respect to apoint in a plane in the X direction and the Y direction, circularsampling can also be performed. This is supported by the fact that nocoefficient of an angle θ is contained in the portion of the weightedexponential function in equation (4) in polar coordinates obtained bysubjecting equation (2) to coordinate transformation.

$\begin{matrix}{z_{({a,b})}^{\prime} = {\frac{1}{N^{2}}{\sum\limits_{\theta}{\sum\limits_{r}{\exp {\left\{ {- \frac{r^{2}}{2\sigma^{2}}} \right\} \cdot z_{({x,y})}}}}}}} & (5)\end{matrix}$

In the present embodiment, referring to FIGS. 6A and 6B, there is noparticular limitation on the best fit plane calculation step S5 as longas it calculates a best fit plane 50 having the minimum distance towarped surface 30. On this occasion, to calculate best fit plane 50having the minimum distance to warped surface 30 means to calculate bestfit plane 50 having the minimum distance to a plurality of pointsrespectively represented by the plurality of displacement values forcalculation subjected to smoothing on warped surface 30. It ispreferable to calculate best fit plane 50 to minimize a sum of squaresof every distance between best fit plane 50 and each point representedby each of the plurality of displacement values for calculationsubjected to smoothing on warped surface 30 (least square method). Withsuch a least square method, best fit plane 50 representing averageinclination of entire rear surface 10 r of substrate 10 supported atthree points can be obtained.

Further, in the present embodiment, referring to FIGS. 6A and 6B, thewarpage calculation step S6 calculates as warpage a sum of a distance D₊from best fit plane 50 to a point z_(p) represented by the greatestdisplacement value of warped surface 30 on one side with respect to bestfit plane 50 and a distance D⁻ from best fit plane 50 to a point z_(v)represented by the greatest displacement value of warped surface 30 onthe other side with respect to best fit plane 50. There is no particularlimitation on such a calculation method. For example, referring to FIG.11, the warpage calculation step S6 in the present embodiment calculatesas warpage a sum of a distance D₊ from best fit plane 50 to point z_(p)represented by the greatest displacement value on one side with respectto best fit plane 50 and a distance D⁻ from best fit plane 50 to pointz_(v) represented by the greatest displacement value on the other sidewith respect to best fit plane 50, in a plurality of points zrespectively represented by the plurality of displacement values forcalculation subjected to smoothing on warped surface 30. In FIG. 11, apoint displaced from best fit plane 50 on one side (including a point onbest fit plane 50) is indicated as a point z_(A), and a point displacedfrom best fit plane 50 on the other side is indicated as a point z_(B).With this manner, the average inclination of entire rear surface 10 r ofsubstrate 10 represented as best fit plane 50 can be compensated forfrom warped surface 30, and the warpage of rear surface 10 r ofsubstrate 10 can be measured accurately.

The direction of warpage can be indicated using a sign. For example,when the rear surface has a concave warped surface 30 as shown in FIG.6A, such warpage is referred to as positive (+) warpage, and when therear surface has a convex warped surface 30 as shown in FIG. 6B, suchwarpage is referred to as negative (−) warpage.

Second Embodiment

Referring to FIGS. 1 to 3, in another embodiment of the method ofmeasuring warpage of a rear surface of a substrate in accordance withthe present invention, a method of measuring warpage of rear surface 10r opposite to crystal growth surface 10 c of substrate 10 using laserdisplacement meter 15 is provided, and substrate 10 is disposed onsubstrate support table 12. The method includes: the substrate detectionstep S1 detecting the plurality of displacement values respectivelycorresponding to the plurality of measurement points 10 p on rearsurface 10 r of substrate 10 using laser displacement meter 15; thenoise removal step S2 removing noise contained in the plurality ofdisplacement values; the outer peripheral portion removal step S3calculating the plurality of displacement values for calculation byremoving from the plurality of displacement values those respectivelycorresponding to the measurement points in the outer peripheral portionof substrate 10; the smoothing step S4 smoothing the plurality ofdisplacement values for calculation to calculate a warped surface; thebest fit plane calculation step S5 calculating a best fit plane havingthe minimum distance to the warped surface; and the warpage calculationstep S6 calculating as warpage a sum of a distance from the best fitplane to a point represented by the greatest displacement value of thewarped surface on one side with respect to the best fit plane and adistance from the best fit plane to a point represented by the greatestdisplacement value of the warped surface on the other side with respectto the best fit plane, wherein an optimization cycle C1 including thesmoothing step S4, the best fit plane calculation step S5, and thewarpage calculation step S6 is repeated one or more times.

By repeating such optimization cycle C1 one or more times, the warpedsurface of rear surface 10 r of substrate 10 can be more smoothed,thereby reducing influence due to surface roughness, and thus thewarpage of rear surface 10 r can be measured more accurately. In orderto measure the warpage of rear surface 10 r more accurately, it ispreferable that optimization cycle C1 includes the smoothing step S4,the best fit plane calculation step S5, and the warpage calculation stepS6 in this order. Further, optimization cycle C1 may include the noiseremoval step S2 after the smoothing step S4.

Although there is no particular limitation on the number of repeatingoptimization cycle C1, the number can be set such that a differencebetween a value of warpage before an optimization cycle and a value ofwarpage after the optimization cycle is preferably not more than 0.5 μm,and more preferably not more than 0.1 μm. Further, the number can be setsuch that a ratio of a difference between a value of warpage before anoptimization cycle and a value of warpage after the optimization cycleto the value of warpage before the optimization cycle is preferably notmore than 0.05, and more preferably not more than 0.01.

Third Embodiment

Referring to FIGS. 1 to 3, in still another embodiment of the method ofmeasuring warpage of a rear surface of a substrate in accordance withthe present invention, a method of measuring warpage of rear surface 10r opposite to crystal growth surface 10 c of substrate 10 using laserdisplacement meter 15 is provided, and substrate 10 is disposed onsubstrate support table 12. The method includes: the substrate detectionstep S1 detecting the plurality of displacement values respectivelycorresponding to the plurality of measurement points 10 p on rearsurface 10 r of substrate 10 using laser displacement meter 15; thenoise removal step S2 removing noise contained in the plurality ofdisplacement values; the outer peripheral portion removal step S3calculating the plurality of displacement values for calculation byremoving from the plurality of displacement values those respectivelycorresponding to the measurement points in the outer peripheral portionof substrate 10; the smoothing step S4 smoothing the plurality ofdisplacement values for calculation to calculate a warped surface; thebest fit plane calculation step S5 calculating a best fit plane havingthe minimum distance to the warped surface; and the warpage calculationstep S6 calculating as warpage a sum of a distance from the best fitplane to a point represented by the greatest displacement value of thewarped surface on one side with respect to the best fit plane and adistance from the best fit plane to a point represented by the greatestdisplacement value of the warped surface on the other side with respectto the best fit plane, wherein optimization cycle C1 including thesmoothing step S4, the best fit plane calculation step S5, and thewarpage calculation step S6 is repeated one or more times. The methodincludes at least one noise removal step S2 in an interval betweenrepeated optimization cycles C1, or after the smoothing step S4 inoptimization cycle C1.

By performing at least one noise removal step S2 in an interval betweenrepeated optimization cycles C1, or after the smoothing step S4 inoptimization cycle C1, noise contained in the plurality of displacementvalues can be removed more effectively, and the warpage of rear surface10 r can be measured more accurately.

According to the first to third embodiments, warpage of a rear surfaceof a substrate can be measured extremely accurately. However,significant efforts are required for processing and analysis of themeasured displacement values. Therefore, in the present invention,warpage of a rear surface of a substrate can be measured quickly andaccurately by simplifying the processing and analysis of the measureddisplacement values as described in the following embodiments.

Fourth Embodiment

Referring to FIGS. 2, 3, and 7, in still another embodiment of themethod of measuring warpage of a rear surface of a substrate inaccordance with the present invention, a method of measuring warpage ofrear surface 10 r opposite to crystal growth surface 10 c of substrate10 using laser displacement meter 15 is provided, and substrate 10 isdisposed on substrate support table 12. The method includes: thesubstrate detection step S1 detecting the plurality of displacementvalues respectively corresponding to the plurality of measurement points10 p on rear surface 10 r of substrate 10 using laser displacement meter15; the best fit plane calculation step S5 calculating a best fit planehaving the minimum distance to a plurality of points respectivelyrepresented by the plurality of displacement values; and the warpagecalculation step S6 calculating as warpage a sum of a distance from thebest fit plane to a point represented by the greatest displacement valueon one side with respect to the best fit plane and a distance from thebest fit plane to a point represented by the greatest displacement valueon the other side with respect to the best fit plane, in the pluralityof points respectively represented by the plurality of displacementvalues.

Specifically, the present embodiment corresponds to a method ofmeasuring warpage of a rear surface of a substrate in which the noiseremoval step S2, the outer peripheral portion removal step S3, and thesmoothing step S4 are omitted from the method of measuring warpage inthe first embodiment including the substrate detection step S1, thenoise removal step S2, the outer peripheral portion removal step S3, thesmoothing step S4, the best fit plane calculation step S5, and thewarpage calculation step S6. Hereinafter, each step will be described.The substrate detection step S1 is the same as that in the firstembodiment.

Referring to FIG. 7, in the present embodiment, the best fit planecalculation step S5 is performed subsequent to the substrate detectionstep S1. Accordingly, the best fit plane calculation step S5 isperformed based on the plurality of displacement values detected in thesubstrate detection step S1. That is, in the best fit plane calculationstep S5 in the present embodiment, a best fit plane having the minimumdistance to a plurality of points respectively represented by theplurality of displacement values is calculated. Although there is noparticular limitation on a method of calculating a best fit plane havingthe minimum distance to the plurality of points respectively representedby the plurality of displacement values, it is preferable to calculate abest fit plane to minimize a sum of squares of every distance betweenthe best fit plane and each point represented by each of the pluralityof displacement values (least square method).

Further, referring to FIG. 7, in the present embodiment, the warpagecalculation step S6 is performed subsequent to the best fit planecalculation step S5 performed subsequent to the substrate detection stepS1. Accordingly, the warpage calculation step S6 is performed based onthe plurality of displacement values detected in the substrate detectionstep S1 and the best fit plane calculated in the best fit planecalculation step S5. That is, referring to FIGS. 10A and 10B, thewarpage calculation step S6 in the present embodiment calculates aswarpage a sum of distance D₊ from best fit plane 50 to point z_(p)represented by the greatest displacement value on one side with respectto best fit plane 50 and distance D⁻ from best fit plane 50 to pointz_(v) represented by the greatest displacement value on the other sidewith respect to best fit plane 50, in the plurality of points zrespectively represented by the plurality of displacement values. InFIGS. 10A and 10B, a point displaced from best fit plane 50 on one side(including a point on best fit plane 50) is indicated as point z_(A),and a point displaced from best fit plane 50 on the other side isindicated as point z_(B). With this manner, the average inclination ofentire rear surface 10 r of substrate 10 represented as best fit plane50 can be compensated for from the plurality of points z respectivelyrepresented by the plurality of displacement values, and the warpage ofrear surface 10 r of substrate 10 can be measured quickly andaccurately.

Fifth Embodiment

Referring to FIG. 8, still another embodiment of the method of measuringwarpage of a rear surface of a substrate in accordance with the presentinvention corresponds to the method of measuring warpage in the fourthembodiment further including after the substrate detection step S1 andbefore the best fit plane calculation step S5: the noise removal step S2removing noise contained in the plurality of displacement values; andthe outer peripheral portion removal step S3 calculating the pluralityof displacement values for calculation by removing from the plurality ofdisplacement values those respectively corresponding to the measurementpoints in the outer peripheral portion of the substrate, and using theplurality of displacement values for calculation as the plurality ofdisplacement values in the best fit plane calculation step S5 and thewarpage calculation step S6.

Specifically, referring to FIGS. 2 to 4 and 8, the method of measuringwarpage in the present embodiment is a method of measuring warpage ofrear surface 10 r opposite to crystal growth surface 10 c of substrate10 using laser displacement meter 15, and substrate 10 is disposed onsubstrate support table 12. The method includes: the substrate detectionstep S1 detecting the plurality of displacement values respectivelycorresponding to the plurality of measurement points 10 p on rearsurface 10 r of substrate 10 using laser displacement meter 15; thenoise removal step S2 removing noise contained in the plurality ofdisplacement values; the outer peripheral portion removal step S3calculating the plurality of displacement values for calculation byremoving from the plurality of displacement values those respectivelycorresponding to the measurement points in the outer peripheral portionof the substrate; the best fit plane calculation step S5 calculating abest fit plane having the minimum distance to a plurality of pointsrespectively represented by the plurality of displacement values forcalculation; and the warpage calculation step S6 calculating as warpagea sum of a distance from the best fit plane to a point represented bythe greatest displacement value on one side with respect to the best fitplane and a distance from the best fit plane to a point represented bythe greatest displacement value on the other side with respect to thebest fit plane, in the plurality of points respectively represented bythe plurality of displacement values for calculation. Accordingly, themethod of measuring warpage in the present embodiment is a method ofmeasuring warpage in which the smoothing step S4 is omitted from themethod of measuring warpage in the first embodiment.

Hereinafter, each step will be described. The substrate detection stepS1, the noise removal step S2, and the outer peripheral portion removalstep S3 are the same as those in the first embodiment. While the presentembodiment and FIG. 8 also describe the case where the outer peripheralportion removal step S3 is performed after the noise removal step S2 asin the first embodiment and FIG. 1, these steps may be performed ininverse order.

Referring to FIG. 8, in the present embodiment, the best fit planecalculation step S5 is performed subsequent to the outer peripheralportion removal step S3. Accordingly, the best fit plane calculationstep S5 is performed based on the plurality of displacement values forcalculation calculated in the outer peripheral portion removal step S3.That is, referring to FIGS. 10A and 10B, in the best fit planecalculation step S5 in the present embodiment, best fit plane 50 havingthe minimum distance to the plurality of points z respectivelyrepresented by the plurality of displacement values for calculation iscalculated. Although there is no particular limitation on a method ofcalculating best fit plane 50 having the minimum distance to theplurality of points z respectively represented by the plurality ofdisplacement values for calculation, it is preferable to calculate bestfit plane 50 to minimize a sum of squares of every distance between bestfit plane 50 and each point represented by each of the plurality ofdisplacement values (least square method).

Further, referring to FIG. 8, in the present embodiment, the warpagecalculation step S6 is performed subsequent to the best fit planecalculation step S5 performed subsequent to the outer peripheral portionremoval step S3. Accordingly, the warpage calculation step S6 isperformed based on the plurality of displacement values for calculationcalculated in the outer peripheral portion removal step S3 and the bestfit plane calculated in the best fit plane calculation step S5. That is,referring to FIGS. 10A and 10B, the warpage calculation step S6 in thepresent embodiment calculates as warpage a sum of distance D₊ from bestfit plane 50 to point z_(p) represented by the greatest displacementvalue on one side with respect to best fit plane 50 and distance D⁻ frombest fit plane 50 to point z_(v) represented by the greatestdisplacement value on the other side with respect to best fit plane 50,in the plurality of points z respectively represented by the pluralityof displacement values for calculation. In FIGS. 10A and 10B, a pointdisplaced from best fit plane 50 on one side (including a point on bestfit plane 50) is indicated as point z_(A), and a point displaced frombest fit plane 50 on the other side is indicated as point z_(B).

With this manner, the average inclination of entire rear surface 10 r ofsubstrate 10 represented as best fit plane 50 can be compensated forfrom the plurality of points z respectively represented by the pluralityof displacement values for calculation, and the warpage of rear surface10 r of substrate 10 can be measured quickly and accurately. Since thenoise removal step S2 and the outer peripheral portion removal step S3are added in the present embodiment compared to the fourth embodiment,the warpage of the rear surface of the substrate can be measured moreaccurately. Further, in the present embodiment, as in the firstembodiment, it is preferable that the substrate is disposed on thesubstrate support table having three supporting portions such that thecrystal growth surface of the substrate is supported by the threesupporting portions, that the substrate detection step S1 is performedby measuring distances between the laser displacement meter and theplurality of measurement points on the rear surface by a laser focustechnique while moving the substrate support table on which thesubstrate is disposed in a two-dimensional direction in a stepwisefashion, and that the noise removal step S2 is performed using a medianfilter.

Sixth Embodiment

Referring to FIG. 9, still another embodiment of the method of measuringwarpage of a rear surface of a substrate in accordance with the presentinvention corresponds to the method of measuring warpage in the fourthembodiment further including after the substrate detection step S1 andbefore the best fit plane calculation step S5: the outer peripheralportion removal step S3 calculating the plurality of displacement valuesfor calculation by removing from the plurality of displacement valuesthose respectively corresponding to the measurement points in the outerperipheral portion of the substrate; and the smoothing step S4 smoothingthe plurality of displacement values for calculation to calculate awarped surface, and using the plurality of points respectivelyrepresented by the plurality of displacement values for calculationsubjected to smoothing on the warped surface as the plurality of pointsrespectively represented by the plurality of displacement values in thebest fit plane calculation step S5 and the warpage calculation step S6.

Specifically, referring to FIGS. 2 to 4 and 9, the method of measuringwarpage in the present embodiment is a method of measuring warpage ofrear surface 10 r opposite to crystal growth surface 10 c of substrate10 using laser displacement meter 15, and substrate 10 is disposed onsubstrate support table 12. The method includes: the substrate detectionstep S1 detecting the plurality of displacement values respectivelycorresponding to the plurality of measurement points 10 p on rearsurface 10 r of substrate 10 using laser displacement meter 15; theouter peripheral portion removal step S3 calculating the plurality ofdisplacement values for calculation by removing from the plurality ofdisplacement values those respectively corresponding to the measurementpoints in the outer peripheral portion of the substrate; the smoothingstep S4 smoothing the plurality of displacement values for calculationto calculate a warped surface; the best fit plane calculation step S5calculating a best fit plane having the minimum distance to a pluralityof points respectively represented by the plurality of displacementvalues for calculation subjected to smoothing on the warped surface; andthe warpage calculation step S6 calculating as warpage a sum of adistance from the best fit plane to a point represented by the greatestdisplacement value on one side with respect to the best fit plane and adistance from the best fit plane to a point represented by the greatestdisplacement value on the other side with respect to the best fit plane,in the plurality of points respectively represented by the plurality ofdisplacement values for calculation subjected to smoothing on the warpedsurface. Accordingly, the method of measuring warpage in the presentembodiment is a method of measuring warpage in which the noise removalstep S2 is omitted from the method of measuring warpage in the firstembodiment.

Hereinafter, each step will be described. The substrate detection stepS1, the outer peripheral portion removal step S3, and the smoothing stepS4 are the same as those in the first embodiment.

Referring to FIG. 9, in the present embodiment, the best fit planecalculation step S5 is performed subsequent to the smoothing step S4.Accordingly, as in the first embodiment, the best fit plane calculationstep S5 is performed based on the plurality of displacement values forcalculation subjected to smoothing on the warped surface calculated inthe smoothing step S4. That is, referring to FIGS. 11A and 11B, in thebest fit plane calculation step S5 in the present embodiment, best fitplane 50 having the minimum distance to the plurality of points zrespectively represented by the plurality of displacement values forcalculation subjected to smoothing on warped surface 30 is calculated.Although there is no particular limitation on a method of calculatingbest fit plane 50 having the minimum distance to the plurality of pointsz respectively represented by the plurality of displacement values forcalculation subjected to smoothing on warped surface 30, it ispreferable to calculate best fit plane 50 to minimize a sum of squaresof every distance between best fit plane 50 and each point representedby each of the plurality of displacement values for calculationsubjected to smoothing on warped surface 30 (least square method).

Further, referring to FIG. 9, in the present embodiment, the warpagecalculation step S6 is performed subsequent to the best fit planecalculation step S5 performed subsequent to the smoothing step S4.Accordingly, the warpage calculation step S6 is performed based on theplurality of displacement values for calculation subjected to smoothingon the warped surface calculated in the smoothing step S4 and the bestfit plane calculated in the best fit plane calculation step S5. That is,referring to FIGS. 11A and 11B, the warpage calculation step S6 in thepresent embodiment calculates as warpage a sum of distance D₊ from bestfit plane 50 to point z_(p) represented by the greatest displacementvalue on one side with respect to best fit plane 50 and distance D⁻ frombest fit plane 50 to point z_(v) represented by the greatestdisplacement value on the other side with respect to best fit plane 50,in the plurality of points z respectively represented by the pluralityof displacement values for calculation subjected to smoothing on warpedsurface 30. In FIGS. 11A and 11B, the point displaced from best fitplane 50 on one side (including a point on best fit plane 50) isindicated as point z_(A), and the point displaced from best fit plane 50on the other side is indicated as point z_(B).

With this manner, the average inclination of entire rear surface 10 r ofsubstrate 10 represented as best fit plane 50 can be compensated forfrom the plurality of points z respectively represented by the pluralityof displacement values for calculation subjected to smoothing on warpedsurface 30, and the warpage of rear surface 10 r of substrate 10 can bemeasured quickly and accurately. Since the outer peripheral portionremoval step S3 and the smoothing step S4 are added in the presentembodiment compared to the fourth embodiment, the warpage of the rearsurface of the substrate can be measured more accurately. Further, inthe present embodiment, as in the first embodiment, it is preferablethat the substrate is disposed on the substrate support table havingthree supporting portions such that the crystal growth surface of thesubstrate is supported by the three supporting portions, that thesubstrate detection step S1 is performed by measuring distances betweenthe laser displacement meter and the plurality of measurement points onthe rear surface by a laser focus technique while moving the substratesupport table on which the substrate is disposed in a two-dimensionaldirection in a stepwise fashion, and that the smoothing step S4 isperformed using a Gaussian filter.

EXAMPLES First Comparative Example

Warpage of a rear surface of a GaN substrate measuring 5.08 cm (2inches) in diameter by 400 μm in thickness and having a surfaceroughness Ra of a crystal growth surface of 1.5 nm and a surfaceroughness Ra of the rear surface of 42 nm was measured using a flatnesstester employing optical interferometry (FT-17 (optical output unit) andFA-200 (analysis unit) manufactured by NIDEK Co., Ltd.). A semiconductorlaser having a laser wavelength of 655 nm was used for the flatnesstester. The measurement points for measuring the displacement valueswere arranged with a pitch of about 100 μm, and displacements at about70650 measurement points were analyzed. The rear surface of the GaNsubstrate had a warpage of 8.5 μm.

The surface roughnesses Ra of the crystal growth surface and the rearsurface of the GaN substrate were calculated by: performing measurementin a range of 750 μm×700 μm using a laser displacement meter employingthe laser focus technique and in a range of 100 μm×80 μm using a 3D-SEM,respectively; sampling a portion having a reference length from aroughness curve arbitrarily specified in each measurement range, in adirection of a mean line of the roughness curve; summing up absolutevalues of deviations from a mean line of the sampled portion to ameasurement curve; and calculating an average for the reference length.

First Example

Warpage of a rear surface of a GaN substrate identical to that in thefirst comparative example was measured using a laser displacement meteremploying the laser focus technique (LT-9010 (laser output unit) andLT-9500 (laser control unit) manufactured by Keyence Corporation), an XYposition controller (CP-500 manufactured by COMS Co., Ltd.), and a dataanalysis apparatus (CA-800 manufactured by COMS Co., Ltd.). A red colorsemiconductor laser having a laser wavelength of 670 nm was used for thelaser displacement meter.

Referring to FIGS. 1 to 3, firstly the GaN substrate (substrate 10) wasdisposed on substrate support table 12 such that the outer peripheralportion of crystal growth surface 10 c thereof was supported by threesupporting portions 12 h. Then, laser displacement meter 15 was used todetect a plurality of displacement values respectively corresponding tothe plurality of measurement points 10 p on rear surface 10 r of the GaNsubstrate (substrate 10) (the substrate detection step S1). On thisoccasion, measurement points 10 p were arranged with pitch P of 700 μm,and a plurality of displacement values respectively corresponding toabout 5000 measurement points 10 p was measured. Next, noise containedin the plurality of displacement values was removed using an8-neighborhood median filter (the noise removal step S2). Thereafter, aplurality of displacement values for calculation was calculated byremoving from the plurality of displacement values those respectivelycorresponding to up to three measurement points inward from outerperiphery 10 e of substrate 10 (the outer peripheral portion removalstep S3).

Then, the plurality of displacement values for calculation was smoothedusing the 8-neighborhood Gaussian filter with σ=5 after normalizationshown in FIG. 5C to calculate a warped surface (the smoothing step S4).Next, a best fit plane was calculated to minimize the sum of squares ofevery distance between the best fit plane and each point represented byeach of the plurality of displacement values for calculation subjectedto smoothing (the best fit plane calculation step S5). Thereafter, thesum of a distance from the best fit plane to a point represented by thegreatest displacement value of the warped surface on one side withrespect to the best fit plane and a distance from the best fit plane toa point represented by the greatest displacement value of the warpedsurface on the other side with respect to the best fit plane wascalculated as warpage (the warpage calculation step S6). The warpagecalculated as described above was 9.0 μm.

Next, noise contained in the plurality of displacement values forcalculation was removed using the 8-neighborhood median filter again(the noise removal step S2). Thereafter, the optimization cycleperforming the smoothing step S4, the best fit plane calculation stepS5, and the warpage calculation step S6 in this order was repeated once.The warpage calculated as described above was 8.5 μm.

Then, the above optimization cycle was repeated once more. The warpagecalculated as described above was 8.3 μm, having a difference of notmore than 0.5 μm from the previously calculated warpage. Therefore, theoptimization cycle was ended, and the rear surface of the substrate wasdetermined to have a warpage of 8.3 μm.

As seen from the comparison between the first example and the firstcomparative example, the warpage obtained by the method of measuringwarpage in accordance with the present invention was almost identical tothe warpage obtained by the measurement using a conventional flatnesstester employing optical interferometry. Thereby, it was confirmed thatwarpage of a rear surface of a substrate can be measured accurately bythe method of measuring warpage in accordance with the presentinvention.

Second Comparative Example

An attempt was made to measure warpage of a rear surface of a GaNsubstrate measuring 5.08 cm (2 inches) in diameter by 400 μm inthickness and having a surface roughness Ra of a crystal growth surfaceof 3 nm and a surface roughness Ra of the rear surface of 57 nm, usingthe same flatness tester employing optical interferometry as that in thefirst comparative example, in a manner similar to the first comparativeexample. However, since the surface to be measured was rough and thusscattered light, interference fringes required to perform themeasurement failed to be obtained. Therefore, the measurement of thewarpage of the rear surface failed to be performed.

Second Example

Warpage of a rear surface of a GaN substrate identical to that in thesecond comparative example was measured in a manner similar to the firstexample. The warpage calculated after the first warpage calculation stepS6 was 10.9 μm. Then, the noise removal step S2 was performed, andthereafter the optimization cycle similar to that in the first examplewas repeated once. The warpage calculated after the second warpagecalculation step S6 was 10.2 μm. Then, the above optimization cycle wasrepeated once more. The warpage calculated after the third warpagecalculation step S6 was 10.0 μm, having a difference of not more than0.5 μm from the previously calculated warpage. Therefore, theoptimization cycle was ended, and the rear surface of the substrate wasdetermined to have a warpage of 10.0 μm.

As seen from the comparison between the second example and the secondcomparative example, in the case where a substrate has a rough rearsurface with a surface roughness Ra of not less than 50 nm, the methodof measuring warpage in accordance with the present invention wascapable of measuring the warpage of the rough rear surface of thesubstrate, whereas the measuring method using a conventional flatnesstester employing optical interferometry failed to measure the warpage ofthe rough rear surface of the substrate.

Third Example

Warpage of a rear surface of a GaN substrate identical to that in thesecond example was measured using a laser displacement meter employingthe laser focus technique (LT-9010 (laser output unit) and LT-9500(laser control unit) manufactured by Keyence Corporation), an XYposition controller (CP-500 manufactured by COMS Co., Ltd.), and a dataanalysis apparatus (CA-800 manufactured by COMS Co., Ltd.). A red colorsemiconductor laser having a laser wavelength of 670 nm was used for thelaser displacement meter.

Referring to FIGS. 2, 3 and 7, firstly the GaN substrate (substrate 10)was disposed on substrate support table 12 such that the outerperipheral portion of crystal growth surface 10 c thereof was supportedby three supporting portions 12 h. Then, laser displacement meter 15 wasused to detect a plurality of displacement values respectivelycorresponding to the plurality of measurement points 10 p on rearsurface 10 r of the GaN substrate (substrate 10) (the substratedetection step S1). On this occasion, measurement points 10 p werearranged with pitch P of 700 μm, and a plurality of displacement valuesrespectively corresponding to about 5000 measurement points 10 p wasmeasured. Next, a best fit plane was calculated to minimize the sum ofsquares of every distance between the best fit plane and each pointrepresented by each of the plurality of displacement values (the bestfit plane calculation step S5). Thereafter, the sum of a distance fromthe best fit plane to a point represented by the greatest displacementvalue on one side with respect to the best fit plane and a distance fromthe best fit plane to a point represented by the greatest displacementvalue on the other side with respect to the best fit plane, in theplurality of points respectively represented by the plurality ofdisplacement values, was calculated as warpage (the warpage calculationstep S6). The warpage calculated as described above was 11.5 μm.

Fourth Example

Warpage of a rear surface of a GaN substrate identical to that in thesecond example was measured as described below. Referring to FIG. 8, thesubstrate detection step S1 was performed as in the third example. Next,noise contained in the plurality of displacement values was removedusing an 8-neighborhood median filter (the noise removal step S2).Thereafter, a plurality of displacement values for calculation wascalculated by removing from the plurality of displacement values thoserespectively corresponding to up to three measurement points inward fromouter periphery 10 e of substrate 10 (the outer peripheral portionremoval step S3). Next, a best fit plane was calculated to minimize thesum of squares of every distance between the best fit plane and eachpoint represented by each of the plurality of displacement values forcalculation (the best fit plane calculation step S5). Thereafter, thesum of a distance from the best fit plane to a point represented by thegreatest displacement value on one side with respect to the best fitplane and a distance from the best fit plane to a point represented bythe greatest displacement value on the other side with respect to thebest fit plane, in the plurality of points respectively represented bythe plurality of displacement values for calculation, was calculated aswarpage (the warpage calculation step S6). The warpage calculated asdescribed above was 11.1 μm.

Fifth Example

Warpage of a rear surface of a GaN substrate identical to that in thesecond example was measured as described below. Referring to FIG. 9, thesubstrate detection step S1 was performed as in the third example.Thereafter, a plurality of displacement values for calculation wascalculated by removing from the plurality of displacement values thoserespectively corresponding to up to three measurement points inward fromouter periphery 10 e of substrate 10 (the outer peripheral portionremoval step S3). Then, the plurality of displacement values forcalculation was smoothed using the 8-neighborhood Gaussian filter withσ=5 after normalization shown in FIG. 5C to calculate a warped surface(the smoothing step S4). Next, a best fit plane was calculated tominimize the sum of squares of every distance between the best fit planeand each point represented by each of the plurality of displacementvalues for calculation subjected to smoothing on the warped surface (thebest fit plane calculation step S5). Thereafter, the sum of a distancefrom the best fit plane to a point represented by the greatestdisplacement value on one side with respect to the best fit plane and adistance from the best fit plane to a point represented by the greatestdisplacement value on the other side with respect to the best fit plane,in the plurality of points respectively represented by the plurality ofdisplacement values for calculation subjected to smoothing on the warpedsurface, was calculated as warpage (the warpage calculation step S6).The warpage calculated as described above was 11.2 μm.

As is clear from the third to fifth examples, it was possible to measurewarpage of a rear surface of a substrate quickly and accurately in asimple and easy way, also in the measuring method including thesubstrate detection step S1, the best fit plane calculation step S5, andthe warpage calculation step S6 (the third example); the measuringmethod of the third example further including the noise removal step S2and the outer peripheral portion removal step S3 (the fourth example);and the measuring method of the third example further including theouter peripheral portion removal step S3 and the smoothing step S4 (thefifth example).

Although the present invention has been described and illustrated indetail, it is clearly understood that the same is by way of illustrationand example only and is not to be taken by way of limitation, the spiritand scope of the present invention being limited only by the terms ofthe appended claims.

1. A method of measuring warpage of a rear surface opposite to a crystalgrowth surface of a substrate using a laser displacement meter, saidsubstrate being disposed on a substrate support table, comprising: asubstrate detection step detecting a plurality of displacement valuesrespectively corresponding to a plurality of measurement points on saidrear surface of said substrate using said laser displacement meter; anoise removal step removing noise contained in said plurality ofdisplacement values; an outer peripheral portion removal stepcalculating a plurality of displacement values for calculation byremoving from said plurality of displacement values those respectivelycorresponding to the measurement points in an outer peripheral portionof said substrate; a smoothing step smoothing said plurality ofdisplacement values for calculation to calculate a warped surface; abest fit plane calculation step calculating a best fit plane having theminimum distance to said warped surface; and a warpage calculation stepcalculating as warpage a sum of a distance from said best fit plane to apoint represented by the greatest displacement value of said warpedsurface on one side with respect to said best fit plane and a distancefrom said best fit plane to a point represented by the greatestdisplacement value of said warped surface on the other side with respectto said best fit plane.
 2. The method of measuring warpage of a rearsurface of a substrate according to claim 1, wherein said substrate isdisposed on said substrate support table having three supportingportions such that said crystal growth surface of said substrate issupported by said three supporting portions.
 3. The method of measuringwarpage of a rear surface of a substrate according to claim 1, whereinsaid substrate detection step is performed by measuring distancesbetween said laser displacement meter and said plurality of measurementpoints on said rear surface by a laser focus technique while moving saidsubstrate support table on which said substrate is disposed in atwo-dimensional direction in a stepwise fashion.
 4. The method ofmeasuring warpage of a rear surface of a substrate according to claim 1,wherein said noise removal step is performed using a median filter. 5.The method of measuring warpage of a rear surface of a substrateaccording to claim 1, wherein said smoothing step is performed using aGaussian filter.
 6. The method of measuring warpage of a rear surface ofa substrate according to claim 1, wherein said best fit planecalculation step is performed by calculating said best fit plane tominimize a sum of squares of every distance between said best fit planeand each point represented by each of said plurality of displacementvalues for calculation subjected to said smoothing.
 7. The method ofmeasuring warpage of a rear surface of a substrate according to claim 1,wherein an optimization cycle including said smoothing step, said bestfit plane calculation step, and said warpage calculation step isrepeated one or more times.
 8. The method of measuring warpage of a rearsurface of a substrate according to claim 7, wherein at least one noiseremoval step is included in an interval between said repeatedoptimization cycles, or after said smoothing step in said optimizationcycle.
 9. A method of measuring warpage of a rear surface opposite to acrystal growth surface of a substrate using a laser displacement meter,said substrate being disposed on a substrate support table, comprising:a substrate detection step detecting a plurality of displacement valuesrespectively corresponding to a plurality of measurement points on saidrear surface of said substrate using said laser displacement meter; abest fit plane calculation step calculating a best fit plane having theminimum distance to a plurality of points respectively represented bysaid plurality of displacement values; and a warpage calculation stepcalculating as warpage a sum of a distance from said best fit plane to apoint represented by the greatest displacement value on one side withrespect to said best fit plane and a distance from said best fit planeto a point represented by the greatest displacement value on the otherside with respect to said best fit plane, in the plurality of pointsrespectively represented by said plurality of displacement values. 10.The method of measuring warpage of a rear surface of a substrateaccording to claim 9, wherein said substrate is disposed on saidsubstrate support table having three supporting portions such that saidcrystal growth surface of said substrate is supported by said threesupporting portions.
 11. The method of measuring warpage of a rearsurface of a substrate according to claim 9, wherein said substratedetection step is performed by measuring distances between said laserdisplacement meter and said plurality of measurement points on said rearsurface by a laser focus technique while moving said substrate supporttable on which said substrate is disposed in a two-dimensional directionin a stepwise fashion.
 12. The method of measuring warpage of a rearsurface of a substrate according to claim 9, further comprising aftersaid substrate detection step and before said best fit plane calculationstep: a noise removal step removing noise contained in said plurality ofdisplacement values; and an outer peripheral portion removal stepcalculating a plurality of displacement values for calculation byremoving from said plurality of displacement values those respectivelycorresponding to the measurement points in an outer peripheral portionof said substrate, and using said plurality of displacement values forcalculation as said plurality of displacement values in said best fitplane calculation step and said warpage calculation step.
 13. The methodof measuring warpage of a rear surface of a substrate according to claim12, wherein said noise removal step is performed using a median filter.14. The method of measuring warpage of a rear surface of a substrateaccording to claim 12, wherein said substrate is disposed on saidsubstrate support table having three supporting portions such that saidcrystal growth surface of said substrate is supported by said threesupporting portions.
 15. The method of measuring warpage of a rearsurface of a substrate according to claim 12, wherein said substratedetection step is performed by measuring distances between said laserdisplacement meter and said plurality of measurement points on said rearsurface by a laser focus technique while moving said substrate supporttable on which said substrate is disposed in a two-dimensional directionin a stepwise fashion.
 16. The method of measuring warpage of a rearsurface of a substrate according to claim 9, further comprising aftersaid substrate detection step and before said best fit plane calculationstep: an outer peripheral portion removal step calculating a pluralityof displacement values for calculation by removing from said pluralityof displacement values those respectively corresponding to themeasurement points in an outer peripheral portion of said substrate; anda smoothing step smoothing said plurality of displacement values forcalculation to calculate a warped surface, and using the plurality ofpoints respectively represented by said plurality of displacement valuesfor calculation subjected to smoothing on said warped surface as theplurality of points respectively represented by said plurality ofdisplacement values in said best fit plane calculation step and saidwarpage calculation step.
 17. The method of measuring warpage of a rearsurface of a substrate according to claim 16, wherein said smoothingstep is performed using a Gaussian filter.
 18. The method of measuringwarpage of a rear surface of a substrate according to claim 16, whereinsaid substrate is disposed on said substrate support table having threesupporting portions such that said crystal growth surface of saidsubstrate is supported by said three supporting portions.
 19. The methodof measuring warpage of a rear surface of a substrate according to claim16, wherein said substrate detection step is performed by measuringdistances between said laser displacement meter and said plurality ofmeasurement points on said rear surface by a laser focus technique whilemoving said substrate support table on which said substrate is disposedin a two-dimensional direction in a stepwise fashion.